# Quadratic Equations in Square Roots

For solving Quadratic Equations in Square Roots, it is required to square both sides entirely, but not individually. For instance,

$$ \begin{align} (1 + 2)^2 &= 3^2 \\

1^2 + 2^2 &\ne 3^2 \end{align} $$

It is important to check the solutions to see if they work for the original equation if the original equation is squared.

### Worked on Examples of Quadratic Equations in Square Roots

Solve \( \sqrt{x+5} + \sqrt{x-2} = \sqrt{5x-6} \) .

\( \begin{aligned} \require{color} \require{AMSsymbols} \displaystyle

\Big(\sqrt{x+5} + \sqrt{x-2}\Big)^2 &= \sqrt{5x-6}^2 &\color{red} \text{square both sides} \\

x+5 + 2 \sqrt{x+5} \sqrt{x-2} + x-2 &= 5x-6 \\

2 \sqrt{x+5} \sqrt{x-2} &= 3x-9 \\

4(x+5)(x-2) &= (3x-9)^2 &\color{red} \text{square both sides} \\

4(x^2 + 3x-10) &= 9x^2-54x + 81 \\

5x^2-66x + 121 &= 0 \\

(5x-11)(x-11) &=0 \\

x = \frac{11}{5} \text{ or } 11

\end{aligned} \)

\(

\text{Check whether the answers are OK, but } x= \displaystyle \frac{11}{5} \text{ does not make the original equation.} \\

\text{So the answer is only } x=11.

\)

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